A System-View Optimal Additional Active Power Control of Wind Turbines for Grid Frequency Support

1 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < A System-V i ew Op timal Additi onal Active Power Control of W ind T urbines for Grid Frequency Suppo rt Yubo Zhang, Student Member, IEEE , Zhiguo H ao, and Songhao Y ang, Member, IEE E , Baohui Zhang, Fello w, IEEE Abstract — Additional active p ower control (AAPC) of wind turbines (WTs) is esse ntial to improve the transient frequenc y stability of low-inertia power systems. Most of the existing research has focused on imitating the frequency response of the synchronous g enerator (SG), known a s virtual inertia control (VIC), but are such control laws optimal for the power systems ? Inspired by this question, this paper proposes an optimal AAPC of WTs to m aximize the frequency nadir post a m ajor power deficit. By decoupling the WT response a nd the frequency dynamics, the optimal fre quency tr ajectory i s solved b ased on the trajector y model, and its u niversality is strictly proven. Then the optimal AAPC of WTs is constructed reversely based o n the average system frequency (ASF ) model with the op timal frequency trajectory as the desired control results. The prop osed method can significantly improve the system frequency nadir. Meanwhile, the event insensitiv ity makes it can be deployed based on the on -line rolling update under a hypothetic disturbance, avoiding the heavy post-event c omputational burd en. Finally, simulation r esults in a two-machine power system and th e IEEE 39 bu s power system verify the effectiveness of the optimal AAPC of WTs. Index Terms — Frequency support, frequency nadir, wind turbine (WT), additional active power control (AAPC) , average system frequency (ASF) model, trajectory optimization model. I. I NTRODUCTI ON IND power plays an indispensable role in t he energy transition and carbon neu tra l for its clean and efficient [1] . As the mainstream form of wind energy u tilization, the large-scale installation of th e grid-followin g type of wind turbines (WTs) significan tly impairs the inertia of power systems. The l ow in ertia tren d implies weak resistan ce to p ower disturbances, which raises a serious concer n for the frequency stab ility of power systems. To improve the frequency stab ility of low-inertia power systems, th e WT is requested to p rovide proper frequency support . Fortunately, vector contro l allows its output power to be rapid ly and precisely adjusted within a reasonable ran ge. With this feature, t he ad ditional active power control (AAPC) is enabled to directly regulate the WT active power in response to the f requency excursion . In g eneral, the existing AAPC ca n be roughly divided into three catego ries, namely the WT -view , the wind farm ( WF)-view, and the system-view m ethods. The WT -view AAPC is fully decentralized , intending to excavate the frequency support ability of the WT itself. The most popular WT -view AAPC meth od is the vi rtual-in ertia control (VIC). The WT with VIC generates an a dditional active power th at depends on the frequen cy excursion, the r ate o f change of f requency (RoCoF ), or both of them . The VIC with fixed parameter s was first proposed in [2], and this pioneering work revealed the potential of the WT in the grid frequency support . Considering the limitation of fixed par ameters to varying scenario s, adaptive VIC attracts extensive attentio n [3- 6]. In [3] and [ 4], the proportional -derivative coefficients of t he VIC were adaptively adjusted according to the WT’s releasable kinetic energy. Using the input-output feedback linearization method, a time -varying controller was designed to provid e an inertial response in [5 ]. Besides , an ad aptive droop con trol of the WT was prop osed [6] . In [7] , both the RoCoF and the WT’s rotor speed were taken into consid eration, aim ing to mitigate the degradation of the VIC under severe contingencies. To avoid the strong dependence on the measured frequen cy and VIC coef ficients, a fast frequency regulatio n (FR) strategy of the WT based o n variable p ower point track ing control was proposed in [ 8] . Overall, the WT -view AAPC of the WT is self- governed, simple and high ly reliab le, but lack s collabo ration with other WTs an d perception of system demands. Inspired b y optimal control such as mod el predictive co ntrol (MPC), the WF-view AAPC strateg ies have b een extensively studied on the coop eration of WTs. In [9], the WF-view power reference was p roperly shared among WTs by solving an MPC problem , o f which the op timization goal is minimizing wind energy loss, etc. Further , the coop eration o f m ultiple WFs was included in the optimization model in [10]. To deal with the inaccuracy of MPC prediction, a double- layer control framework was pro posed to ad equately exploit the kin etic energy of all WTs [11] . In [ 12], the data -driven MPC method was p roposed to cop e with the challen ge of comp lex and nonlinear WT dynamics. In the aspect of improv ing the computation al efficiency, the distributed Newto n method endowed the proposed method with a super-linear convergence rate in [1 3]; b esides, the off -line look -up tab le was m entioned in [14] and WTs clustering s imp lification was proposed in [15] . The system-view strategies expect to explore the feasible AAPC o f WTs fr om the perspective of system freq uency optimization. Fo r example, the particle swarm algo rithm was applied in [16] to derive the op timal FR con trol p arameter o f the WT with the goal of improving the frequency nadir as well as the average frequency. However, the optimizatio n results were event-sensitive to some e xtent . In [17], a multi-layer M PC was p roposed to achieve both objectives of dynamically optimal power support of the WF and the stability of WTs. Despite the improv ed performance, MPC may unduly consume W This work was support ed by National Natural Science Founda tion of China (52007143) , China Postdoct oral Science Foun dation (2021M69252 6) and K ey Research and De velopment Pr ogram of Shaanxi(2022GXLH- 01 - 06 ). Y. Zhang, Z. Hao, S. Yang and B. Zhang are with State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi'an, China (e-mail: zyb97 0305@stu.xjtu.e du.cn, {zhghao, song haoyang, bhzhang}@xjtu. edu.cn). 2 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < the p ower support capacity of WTs in a short time. An op timal synthetic inertia control considering the variation of mechanical power of th e WT is proposed in [18] , which avoids th e secondary drop of sy stem frequency while satisfying the preset frequency nadir constraint. To minimize the RoCoF of power systems, ref. [19] presented a s ynthetic inertia controller for t he WT based on the 2 H optimal control method. However, the mechanical power of SGs was req uired to be measured in real- time, which is difficult in practice. To further exploit the potential of WT s on improving system frequency stability , th is paper pr oposes a sys tem-view optimal AAPC of WTs . The m ain contributions o f this work include: 1) The o ptimal frequen cy trajectory that m aximizes the frequency nadir is giv en and its universality is strictly proved. By decoupling the WT response and the frequency dynam ics, th e optimal frequen cy trajector y is efficiently solved by the trajectory o ptimization model. Furthermor e, th e univ ersality of the optimal frequency trajectory is strictly proved from the perspective of energy, r egardless of the system p arameters and the disturbance m agnitudes. 2) The feedback-for m optimal AAPC of WTs is derived from the ASF model to ac hieve the optimal frequency trajectory. In addition , an ex it strateg y is design ed as a supplement to the optimal AAPC to ensure the complete state recovery of the WT. 3) The proposed method possesses the following two main advantages . In terms of control effect s, the optim al AAPC of WTs m aximizes the system freq uency nadir. It adaptively responds to power system deman ds rather than blindly grabbing the WT’s frequen cy support capability. In terms of application, e vent insensitivity makes the optimal AAPC of WTs can be solved an d deployed based on the on-line rolling update under a hypothetic disturbance, which a voids the heavy post -event computation al burden. The rest of the paper is o rganized as fo llows. Section II briefly introduces the system frequency and the WT model. Section III shows the design process of the optimal AAPC. The pr oof o f the universality of the optim al AAPC is exh ibited in Sectio n IV. Then, the performance of th e optimal AAPC is v erified in Section V and Section VI. Finally , Section V II draws a conclusion. II. S YSTEM F REQUEN CY AND WT M ODELING A. Dynamic Model of Power System Frequen cy When the power system suffers a power d isturbance, the system frequency will de viate from the nominal value and undergo a transient process. Considering the frequency support of WTs, the freq uency dynamic p osts a larg e d isturbance can be modeled as th e following swing equ ation () 2 ( ) ( ) ( ) m e d d f t H P t P t P D f t dt  =  +  − −  , (1) where H and D denote the normalized inertia constant and damping constant; () ft  is the f requency excursion; m P  and e P  are the FR power of SGs and that of WTs, respectively ; d P Fig. 1 ASF model c onsidering the AA PC of WTs. is th e p ower disturbance . Considering th at the frequency d rop caused by a majo r p ower deficit is m ore common and severe, this paper fo cuses on the scenario of a frequen cy drop. During the p rimary fr equency regulatio n (PFR), generato r s’ governors and AAPC of WTs actively respond to the power imbalance. This process can be described using an ave rage system frequency (ASF) model [20], as shown in Fig. 1. As for the sym bol in Fig. 1, g and w denote th e set o f traditional SGs and WT s, respectively. () gi Gs is the F R dynamics th e governor; () wj Gs is the AAPC of th e WT. Th e sub script “ i ” denotes the i - th SG, and the subscript “ j ” denotes the j -th WT, the same as belo w. In this paper, WTs are supposed to operate in the maximum power po int tracking (MPPT) mod e and only provide short- term frequen cy support . Hence , the steady -state frequency excursion of the PFR is no t affect ed . Acc ording to [ 21], it can be calculated as d ss g P f DK   = − + (2) wh ere () g g gi gi b i K P R S  =  and gi P is the rated power of the SG ; 1 gi R is the steady -state gain factor of the governor of the SG. Nevertheless, the transient p erformance of PFR can be effectively improved by optimizin g the AAPC of WT s, wh ich has bee n co nfirmed in exten sive research. As one of the most significant indicators of PFR, the frequency nadir ref lects the worst tr ansient freq uency drop post a power d eficit. Therefore, this paper focu se s on exploring the optimal AAPC to maximize the frequency nadir. B. WT model In this pap er, one of the most widely-u sed variab les spee d WT is co nsidered, namely the doubly-f ed induction generator (DFIG) WT. The turbine p ower of a WT is calculated b y t he following mod el. 23 0.5 ( , ) t p w P R C v    = (3) 3 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < where  is th e air d ensity; R is t he turbine radius; w v is the wind sp eed; rw Rv  = is the tip-spee d ratio; r  is the rotor speed;  is the pitch angle; ( , ) p C  is the power coefficient, which is related to the tip -speed ratio and pitch angle. The widely used tu rbine efficiency model is adopted in th is paper. 12.5 3 116 ( , ) 0.22 ( 0.4 5) , 1 0.03 5 0.08 1 p Ce         − = − − =− + + (4) The mechanical dyn amics of the WT depend on its turbine power an d output power. According to [22], th e one-mass model is adeq uate to investigat e the WT dynamics duri ng the frequency suppo rt, as follows. r r t e J P P  =− (5) where J is the inertia of the motion system of the WT. C . Constraints in Frequ ency Suppo rt of WT s The short-term f requency support of WT s is essentially to inject excess active power into the system at the expense of releasing the kinetic energy of the rotor. Inevitably, this additional active power will ca use the speed of WTs to decrease and deviate from the steady st ate. Hence , the released energy is required to be injected back in to WTs for the rotor speed recovery. Ignoring the variation of the turbine p ower, the energy absorption char acteristic of WTs is modeled as follows. 0 ( ) 0 f t e t P t dt =  (6) where 0 t and f t are the start time and th e en d time of the freq ue ncy support of WTs, respectiv ely. For simplicity, the start time 0 t is set to 0. To properly exploit th e frequency support potential of WTs as well as en sure their safe operation , the end time of frequen cy su pport is recommen ded as 30s, which corresponds to th e time scale of the PFR. In ad dition, op erating constraints such as the ou tp ut power and the ro tor speed unsurpr isingly affect the frequen cy support performance of WT s. However, these oper ational co nstraints are generally event-sensitive. That is to s ay, whether c onstraints work or not is determin ed by the severity o f the disturbance. Given the unpredictability o f the d isturbance, this p aper expects to explore a more general and event-insensitive AAPC of WTs , thus av oiding h eavy computation . Th erefore, these event- sensitive constrain ts o f the WT ar e igno red when deriving the optimal AAPC , and only t he “ energy absorption” characteristic is retained . No tably, the operating co nstraints of the WT are formally ignored in the theoretical derivation, which are completely r etained in the simulation model. III . O PTIMAL AAPC D ESIG N For the high -order ASF mod el shown in Fig . 1, it is intractab le to directly derive the optimal AAPC of WT s that maximizes the freq ue ncy nadir. Hen ce, this paper explores the optimal AAPC of WT s based on rev erse thinking. By decoupling the WTs response and frequency dynamics, the theoretically optimal f re quency trajectory is fr eely explored. Then, the optimal AAPC is reversely co nstructed from the ASF model with the optimal frequen cy trajectory as the desi red control r esult. Fig. 2 Framework of the optimal AAP C design. The specific design framework is shown in Fig. 2. Firstly, the closed loop in the ASF model is b roken and an aggreg ate additional power is introduced for dec oupling the calculation of WT s respon se and freq uency dynam ics. Then, taking the additional power of WT s as the control variab le, a trajectory optimization model is constructed to m aximize th e freq uency nadir. Based on the time- domain solution of the tr ajecto ry optimization model, the optimal AAPC of WTs can b e reversely constructed fro m t he ASF mo del . Finally, considering the mo del error caused by the ignoran ce of the variation of turbine po wer of WTs, a proper exit strategy is proposed as a supplemen t to ensure the sp eed recovery of WTs. A. Breaking th e Closed Loop of the A SF Model For the hig h-order ASF model shown in Fig. 1, it is d ifficult to derive its analytical d ynamic in the time domain. Meanwhile, the freq uency support o f WT s is cou pled with the frequency , which cau ses an obscure impact o f AAPC on the freq uency nadir . Th e complex coupling and high-order prope rties make it challenging to directly solv e the optimal AAPC of WT s . Generally , WT s are co nsidered to b e able to o utput any reasonably desire d power curve giv en th eir excellent controllability . Th erefore, this pap er simplifies th e pr oblem by breaking the closed loop of the ASF mo del. Spec ifically, the additional power of all WT s is ag gregated as an independent controllable variab le. As an independent control variable, the aggregated additional power o f WTs exerts a unidirec tional effect on the frequency, which makes it p ossible to freely explore the optimal co ntrol curve of WTs that maxim izes the frequency n adir. B . Trajectory Optimization Model Taking the aggregate additional power of all WT s as th e independent control variable, the trajectory o ptimization model can b e constructed to solve the optimal control curve of WTs and the corresponding frequency trajectory in the time domain [23]. Different from the MPC wh ich nee ds to solve th e optimization model in real- time to generate single -step control, the trajectory optimization model der ives the optimal con trol curve in the entire period at once. So, the trajectory optimization 4 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < model do es not require real -time calculation. The construction of the trajecto ry optimization model is introduced below. First, a new state variable associated with the released energy of WT s is introdu ced in (7), aiming to convert the integral constraint shown in (6) into scop e constraints. 0 ( ) ( ), [ , ] e f E t P t t t t  =   (7) Then, the integral constraint in (6) can be converted into 0 ( ) ( ) 0 f E t E t  −  = (8) The dynamic constraints consist of the swing equation, governor dynamics, an d Eq. (7 ), which can be expressed as follows. 1 2 0 ( ) ( ) ( ) , [ , ] e d f t t P t P t t t = +  +  x A x B B (9) where, () ( ) ( ) () g ft tt Et    =     xx , ( ) 2 2 0 0 0 0 gg gg D D H H −   =    C A B A 0 , 1 12 1 H   =    B0 , 2 12 0 H −   =    B0 , () e Pt  is the control variab le ; g A , g B , g C , g D , an d g x model the dyn amics of governors , see Appendix A for details. Since the power system is supposed to operate at the steady- state point initially, the initial values of all the state variables in (9) are 0. Co mbined with (8), the en dpoint constraints o f the trajectory optimizatio n model are as follows. 00 0 ( ) ( ) ( ) 0 () f g f t E t E t t  =  =  =    =   x0 ( 10 ) A static variable nadir f  is introduced, denoting the deviation of the frequency nadir relative to th e base frequen cy. nadir nadir B f f f  = − ( 11 ) where nadir f is th e frequen cy n adir in the tran sient process of PFR ; B f is the base valu e of the system frequency. Naturally, the frequency variation should no t exceed this v alue in the transien t process. 0 ( ) , [ , ] nad ir f f t f t t t     ( 12 ) where () ft  and nadir f  are both negative. Finally, the following optimization objective is designed from a system perspectiv e , aiming to imp rove the frequency nad ir. max e nadir P f   ( 13 ) Based on the above processing , the fo llowing trajectory optimization model can be constructed. Specifically, Eq. ( 13 ) is the optimization ob jective ; Eq. ( 9) is th e dy namic constraint ; Eq. ( 12 ) is the p ath constraint; Eq. ( 10 ) is the endpoint condition s. Th en, th e o ptimal co ntrol curv e and f requency trajectory in the time domain can be solved efficiently b y the Gauss pseu dospectral method [ 24], which is brief ly introduced in Append ix B. The solution of th e trajectory o ptimization model corresponding to th e two-m achine system in Section V is shown in Fig. 3 . Sp ecifically, Fig. 3(a) shows the aggr egate additional Fig. 3 Time-domain solution of trajectory optimization model. (a) Optimal additional power of WTs. (b) Release d energy of WTs. (c) Optimal freque ncy dynamics. (d) Traje ctory of the equiv alent WT. power of WTs, which is positi ve in the initial stage to sup press the fr equency drop, and negative in the later p rocess for rotor speed recovery. Accordingly, the kinetic energy is released initially and then entirely absorbed from the grid, as shown in Fig. 3(b). The frequency dy namics shown in Fig. 3(c) indicate the optim al control of WTs main tains the frequency at a constant nadir, denoted as nadir f   , even in the process of energy absorption . Fig. 3(d) f urther shows the trajectory of the equivalent WT. Ignoring the variation of turbine power, WTs theoretically recov er to the steady -state operation at the end of frequency support, which is guaranteed by the constraint in (6). C . Feedback Approximation The optimal frequency trajectory in the time domain is obtained by solving the trajectory optimizatio n model . Then , the optimal feed back-form AAPC of WTs is derived based on reverse approx imation, as follows. By observing the shape of the optimal frequency trajec tory in Fig. 3(c) , th e fr equency d ynamics can be approximated as th e following exponential decay f orm, as shown in Fig. 4 . ( ) ( 1 ) bt f t a e −  = − ( 14 ) When t → , the frequency should be equal to nadir f   . So the coefficient a can be d erived as follows. d nadir ss g P a f f DK    =  =  = − + ( 15 ) where nad ir ss ff   =   . Moreover, the initial RoCoF is calculated as 0 2 d t d f P ab dt H =  = = − ( 16 ) Hence, the co efficient b can be derived as 2 g DK b H  + = ( 17 ) Then the s -domain frequency dynam ics can be expressed as () (2 ) d g P fs s Hs D K    = − ++ ( 18 ) 5 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < Fig. 4 Theoretic al frequency dynami cs and its exponential approximation. Substituting the s -domain frequency dy namics into the ASF model shown in Fig . 1, the aggregated optimal AAPC in the feedback for m can be derived as follows. ( ) ( ) w g w G s G s K = − + ( 19 ) where ( ) ( ) g g gi i G s G s  =  , () wg K D D K  = − + . The expression of the o ptimal AAPC of WTs is free from disturbance d P . Hence, the optimal AAPC for a given power system is indep endent of the magn itude of the power deficit , which is claimed as event insensitivity. Eq.( 19 ) fur ther indicates that the optimal AAPC of WTs consists of two p arts. The part () g Gs − makes WTs produce a po wer com ponent opposite to the regulation of the mec hanical power of SGs. From the system per spective, this power componen t simplifie s the system frequen cy model to be first -order, thus eliminating the conventionally convex frequency nadir. On the other hand, according to the features of the g overnor of SGs, this po wer component is neg ative, which cond uces to the speed recovery from the WT perspective. Another part w K produ ces a power component th at increases linearly with the frequen cy variation, thereby adjusting the value of the frequency nad ir. In the multi -WTs system, the aggregated optimal AAPC needs to be reason ably allocated to each WT, as follows. ( ) ( ), 1 w wj j w w j j G s c G s j c  =    =    ( 20 ) where j c is the allocation factor of the WT . In this pap er, an allocation strategy considering bo th the releasable kinetic energy and the increasable active power of the WT is propo sed, with details as shown in Ap pendix C. Therefore, the output power o f the WT durin g the frequency support period of 0 [ , ] f t t t  is calculated as 0 ( ) ( ) , ej ej wj j w P P G s f s j = +   ( 21 ) wh ere 0 ej P is th e stead y-state outp ut p ower of the WT bef or e the power deficit disturbance; j f  is the local frequency of the WT. It indicates that the proposed optimal AAPC can be deployed locally to WTs like the classic VIC, which shows the good applicatio n potential of the optimal AAPC. Moreover, it shou ld be emph asized that the consistency between the optimal AAPC and the time-dom ain control relies on th e exponential decay form of ( 14 ) is a goo d approximation of th e optimal frequency trajec tory. That is to say, the shape of the op timal frequency dynamics shown in Fig .3 (c) sho uld be universal for the trajec tory optimization model of different power systems an d disturbance m agnitude. Th e p roof of the required univ ersality is given in Sectio n IV. D. Exit Strategy of Frequency Support Du e to the coupling of rotor speed and turbine power, the WT is usually unable to recover to th e steady state at the en d of the optimal AAPC . In other words, there is a certain deviation between the trajectory of W T derived from the constant tu rbine power and the actual trajectory. T o ensure the state recovery of the WT , th e following exit strategy is designed . For any WT regulated by the o ptimal AAPC, the fo llowing three even ts will trigger the exit strategy . m in ( ) ( ) ( ) ej MP P Tj f rj r P t P t t t t     or or ( 22 ) where w j  ; MPPTj P denotes the MPPT power of the WT; rj  denotes the rotor speed of t he W T; min r  denotes the lower limit of rotor speed of the WT. Denote the activation momen t of the ex it strateg y as ej t , the output po wer of the WT for the ex it strategy is design ed as ( ) ( 1 ) ( ) ( ), ej j tj j MPPTj ej P t P t P t t t  = − +  ( 23 ) where j  is a coefficien t with a value range of [0,1 ] j   . T o eliminate the power saltation caused by control switching, the coefficien t j  is designed as follows: substituting o utput power o f the WT at the ej tt = into ( 23 ) to deriv e the valu e of j  , if it exceeds th e limits, take boun dary value. ( ) ( ) ( ) ( ) ej ej tj ej j MPP Tj ej tj ej P t P t P t P t  − = − ( 24 ) IV. U NIVERSALI TY O F O PTIMAL F REQUENCY T RAJECTORY As mentioned above, it is necessary to prove that the shape of the optimal frequency trajectory in Fig. 3(c) is general. Hence, the proof is given as follows. Integrating the swing equation at the time interval 0 [ , ] f t t t  and combinin g the constraints shown in (6), one obtains () 22 m d f ff D E P t f t S HH  −  + = , ( 25 ) where 0 () f t f t S f t dt  =  , 0 () f t mm t E P t dt =  . m E is the FR en ergy of SGs at the time in terval 0 [ , ] f t t t  ; () f ft  is the terminal f requency at f tt = . Theorem 1: Optimal AAPC of WTs must maximize th e FR energy of SGs. Proof: Assuming th e FR energy of SGs is not eq ual under two different AAPC o f WTs, it is advisable to set 12 0 mm EE  . ( 26 ) In the power deficit cases, both sides of ( 25 ) are less than zero, then one obtain s 21 0 22 m d f m d f E P t E P t HH  −− = ( 27 ) wher e 01   . Denote the fr equency at the time in terval 0 [ , ] f t t t  under 1 m E and 2 m E as 1 () ft  and 2 () ft  , respectively. One obtains 2 2 1 1 ( ) ( ( ) ) 22 f f f f DD f t S f t S HH    + =  + ( 28 ) 6 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < Fig. 5 Frequency trajectory chara cteristic. An obvious case satisfy ing the above equation is 2 1 0 ( ) ( ), [ , ] f f t f t t t t   =   ( 29 ) Therefore, the frequency nadir satisfies 2 1 1 nadir na dir nadir f f f   =    . ( 30 ) The above analysis sho ws that there certain ly ex ists a higher frequency nadir under the larger FR energy of SGs. Therefore, the optimal AAPC o f WTs mu st maximize the FR energy of SGs, which is an u nknown con stant under a ce rtain power deficit, den oted as max m E . Theorem 2 : The frequ ency n adir can no t be lower than the terminal frequen cy () f ft  . Proof: It can be proved by contradictio n. Suppose there is a frequency nadir below the terminal fr equency, as shown in the blue solid line in Fig. 5. () nadir f f f t    ( 31 ) According to Theorem 1 , the right -hand side of ( 25 ) is a constant with m ax mm EE = , denoted as C . Therefore, the o ptimal frequency dyn amics satisfy ( ) ( 2 ) ff f t D H S C   + = ( 32 ) where ma x ( ) 2 m d f C E P t H =− . Particularly, th e envelope dynamics of the frequen cy refers that the frequen cy drops rap idly to th e f requency n adir nadir f  with the initial RoCoF and then maintain s until terminal time, as shown in the g ray solid line in Fig. 5. Co rrespondingly, the integral of this freq uency dy namics is denoted as f S  . Therefore, for any frequency trajectory with the c har acteristics shown in ( 31 ), one can o btain ( ) ( 2 ) ( 2 ) f f nad ir f C f t D H S f D H S C  =  +   + = , ( 33 ) CC  = , ( 34 ) where 01   as C and C are both less than 0; the integral of 0.5 ( ) f f c nadir S t t f  = +  , c f x t t t =− ; x t is th e time to reach the nadir of th e envelope frequency trajectory . For any (0,1 )   , the f ollowing frequency dynamics with a higher frequency nadir can be easily obtained within the envelope trajectory, which satisfies the co nstraint of ( 32 ) . Specifically, the super ior frequency dynamics hav e the same shape as the envelope frequency trajectory, as shown in the red solid line in Fig. 5. Thus, the high er frequency nadir is calculated as nadir nadir ff    =  ( 35 ) According to ( 34 ), th e scale factor  can be d erived as 2 2 2 2 2 2 22 44 2 44 () f f f f c f c fc H D t HDt D t H Dt D t HDt HDt D t t      + + − +− + − + = − ( 36 ) Then, th e value range of  is derived as (0,1 )   2 2 2 (2 ) 0 () 2 (2 ) 1 () ff fc fe fc H Dt H Dt X D t t H Dt H Dt Y D t t    + − + −  = −   + − + +  =  −  ( 37 ) where 2 2 2 ( ) 4 ( ) 0 c f c f X D t t H D t t  = − + −  2 2 2 ( 1 ) ( ) 4( 1 ) ( ) 0 c f c f Y D t t DH t t  = − − + − −  . Therefore, there is always a high er f requency nadir than the primary nadir if it exceed s the terminal frequency, which indicates that th e optim al frequency nadir should not exceed th e terminal frequen cy. Based on Theorem 1 and 2 , t he f requency nad ir maxim ization is equivalent to minimizing the freq uency integral. ma x max ( ) min nadir f f f f t S      ( 38 ) Obviously, the minimum frequ ency integ ral is that the frequency drops rapidly to the frequen cy nadir with initial RoCoF an d then maintains to the term inal time o f f tt = , as shown in Fig. 3(c). The above analysis does not presuppose any information about power systems and d isturbances, so the conclusion is universal. V. C ASE S TUDY I: A T WO - MACHI NE P OWER S YSTEM A tw o- ma ch i ne po we r s ys t em co ns is t in g of an SG a nd a W T is fi r st c on st ru ct ed t o va li dat e t he pr op os e d opt im al A APC , as s ho wn in Fi g . 6 . Th e t ot al ac ti v e l oa d i s 1 50 MW . The W T is a n ag gre ga t e mo de l co nsi st i ng of 20 5MW  D FI G- ba s ed WTs wit h pa r am et er s sh own i n TAB LE I, a nd t he det ai l e d m od el ca n ref e r t o [2 5] . G 1 is th e st ea m tur bi n e wit h a si mp li fi e d go ve rn or mo de l a s s ho wn i n ( 39 ) , a nd t he k e y p a ram e te rs a r e l i st ed in T ABL E I I. ( 1 ) () ( 1 ) m H R g R K F T s Gs R T s + =− + ( 39 ) B as ed o n t he s ys te m pa ra me te rs and t he hy po t het i c pow e r di st u rb an ce of 0. 1 dL PP = , i . e. 15 MW , t he tr a je ct or y o pt im i za ti on mo de l is co ns t ruc t ed an d so lv ed , de r iv in g th e rat io fac t or of 1.186  = . Th e n th e val u e of w K i n the opt i mal AAP C of the WF is cal cu la te d to be 14.1 − . Co mbi n ed wi t h () g Gs , th e opt im al AA PC of t he W T is ob ta in ed as ( ) ( ) w g w G s G s K = − + . A. Properties of the Optimal AAPC B as ed on t he ab ov e tw o -m ac hi ne sys t em , t hi s s ubs e ct io n r ev ea ls th e ess e nt ia l pro pe rt ie s of th e opt i ma l AA PC . The wi nd sp ee d of th e WT is 9 m/ s in t hi s s u bs ec ti on . The p ow er di s tu rb an ce is s et a s an act iv e l oa d sur g e of 15 MW , and the out pu t ch ar ac te ri st i cs o f the pr op os ed o pt im al AA PC a re s how n in Fi g . 7. T he g ree n so li d li ne in Fig . 7 s ho ws th e ad di ti o nal pow er c omp on e nt prod uc e d by the te r m () g Gs − , whi c h i s n e gat i ve a nd exa c tl y of fs ets t he i nc re me nt 7 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < in t he me ch an ic al p owe r of G1 . I n c on t ra st , the te r m w K p ro d uc es Fig. 6 Two-machine power system. TABLE I P ARAMETERS OF THE W IND T URBINE Sym bol Description Value n S Rated capacity ( MVA) 5.556 n P Rated power (MW) 5 max e P Maximum power ( MW) 5 min e P Minimum power ( MW) 0 n  Rated rotor spee d (rpm) 12.1 min r  Minimum rotor s peed (p.u.) 0.7 J Rotor inertia ( 2 kg m  ) 16,801,544 TABLE II P ARAMETERS OF THE S YNCHRONOUS G ENERATOR Symbol Description G1 n S Rat ed capac ity (MVA) 200 m K mechanical power gain factor 0.85 g H Inertia time co nstant (s) 4.2 R governor regu lation 0.05 H F Fraction of high-press ure turbine 0.3 R T Re heat time c onstant (s) 8 a po si ti v e po we r co mp on en t to su pp or t the sy st em fre qu en c y, as s ho wn b y the blu e sol i d li ne in Fi g. 7. Th es e tw o op po si te co mp o ne nt s ca us e th e ove r al l add it i on al pow e r of the W T t o in c re as e fi rs t fo r gr id fre q uen cy s up po rt and the n de cr ea se for s pe ed r ec o ver y , as s ho wn b y t he r ed s o li d li n e i n F ig . 7. F ig . 8 fu rt he r c omp ar es th e t heo re t ic al ly tim e -d oma i n sol u ti on of t he tr aj ec t or y opt im iz at i on mod el and th e sim ul at i on res u lt s of th e pr o pos e d opt i ma l AAP C. As a fee db a ck ap pr ox i ma ti on , th e ou t pu t pow er dyn a mi c of t he pro po se d AAPC is co ns is te nt wi t h th e op ti m al t im e- do ma in c on tr ol c ur ve d er iv e d f ro m th e tr aj ec to ry op t imi z at i on mod el , as sho w n i n Fi g. 8( a) . Co rr es p on di ng ly , th e s ys te m fre qu e nc y pre se nt s th e ex pe c te d fir s t- or der s ha pe , and th e fr e qu en cy nad i r is t he sa m e as th at of th e tr aj ec t or y opt i mi za ti on mo de l , as sh ow n in Fig . 8( b) . As the pr op os ed AAP C in cl u des a po we r co mp on en t tha t pr om ote s rot o r spe e d rec ov er y, t he rot or s pe ed fi rs t dec re as e s a nd the n r ec ov e rs , as s ho wn in Fi g. 8(c ). Ho w eve r , si nce th e t ra je ct o ry op ti mi z at io n mo de l ig nor es win d en e rg y lo ss es , th e ac tu al r ot o r s pe ed doe s no t re co ve r to t he s te a dy- s ta te val u e as ex pe ct ed in the or et ic a l re s ult s . Th is is why a pr op er ex i t st ra te g y i s n ee de d. B. Performance o f the Exit Strategy The exit strategy aims to ensure the state recovery of the WT without deteriorating the frequency nadir. This sub section verifies the p erformance of the p roposed exit strateg y, and the MPPT-based ex it strategy in [16 ] is used as a comparison . Admittedly, the MPPT- based exit strateg y eliminates p ower saltation if the output power curve intersects the MPPT cur ve. Fig. 7 Active power output characteri stics of the optimal AAPC of WTs. Fig. 8 Comparison of pr oposed AAPC and trajectory optimization model. (a)Power su pport of the WF. (b) System fr equency. (c) R otor speed of the W F. However, this case is not inevitable . Fig. 9 and Fig. 10 show the comparison between the proposed exit strategy and th e MPPT- based exit strategy in the above two scenarios. Fig. 9 indicates that both the p roposed exit strategy an d MPPT -b ased m ethod can smooth ly switch to the MPPT mode for speed rec overy when the outp ut power curve intersects with th e MPPT curve. However, when the output power curve do es not intersect with the MPPT curv e, th e MPPT -based exit strategy will ca use th e output power of the WT to plummet to the MPPT curv e at the moment of activ ation, as shown by the blue dotted line in Fig. 10 (a). As a result, th is po wer plu mmet causes a secondary frequency drop, as shown by the blue dotted line in Fig. 10 (b). In contrast, the p roposed exit s trategy maintains the smoothness of the output power wh ile achieving spee d recovery as shown by th e oran ge solid line in Fig. 10 . To conclude, the proposed exit strategy is eff ective and ensures the stable o peration of the 8 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < Fig. 9 Validation of exit strategy with intersection point. (a) Output power of the WT. (b) Gri d frequency dynam ics. Fig. 10 Validation of ex it strategy without int ersection point. (a) O utput power of the WT. (b) Grid frequency d ynamics. WF without deter iorating the frequency n adir. C. Event Insen sitivity As ana l yz ed in Sec ti o n III .C , th e opt i mal AAPC fo r a gi ve n po we r sys te m i s t heo re t ic al ly fre e fr o m th e ma g nit u de of t he po we r de f ic it , nam el y ev ent ins en si t ivi t y. Fi g. 11 sh ow s t he fr eq ue nc y dy na m ic s un der var i ous powe r dis tur ba nc es , an d the opt im al AA PC is de r iv ed fr om a h yp oth et ic po we r de fi ci t of 0.1 dL PP = . Th e r ef e rre d f re qu en cy na di r ca n be s olve d b as e d on t he t ra je ct or y op t imi z at i on mod el , den ot ed as nadir f   . The re su lt s ind ic a te th at th e op ti ma l AA PC ma ke s th e gri d fr e qu enc y ex hi bi t th e de sir e d fi r st -o rd er dyn a mic s reg ar dl es s of th e di st ur ba n ce siz e. The co nve x fr e qu en cy nadi r i s e li mi n at ed an d th e fre qu e ncy nad ir ca n be im p ro ve d as th e ide al so l uti o n of t h e tr aj e ct or y o pt imi z at i on mo del . C ons i de ri ng t he op er at in g con st ra i nt s of the ac t ual WT , th e ev e nt -i ns en si ti v e ra ng e of th e op t im al AAP C s ho ul d be li mi ted . Ta k in g th e fr e que nc y nadi r as t he i nd ic at or , Fi g. 12 furt h er t es ts th e op t im al A AP C un d er a s er ie s o f p ow e r de fi ci t s. T he re su lt s s how Fig. 11 Freque ncy dynamics in t he event-insensitive ran ge of optimal AAP C. Fig. 12 Validat ion of the event insensi tivity of the opt imal AAPC. Fig. 13 Impact of wind speed of the WT on the event-insensitive range o f the optimal AAPC. th a t w it hi n a ce r ta in ran ge of pow e r dis t ur ba nc e , the opt im al AAP C ca n imp ro ve the fre q ue nc y na di r a s ex pe ct e d. H ow ev er , i f th e FR de ma n d und er the ex tr em el y sev er e pow er def i ci t sc en ar io s tha t ex c ee d t he f re qu en cy s up por t cap a bil i ty o f th e WT , th e pe r fo rm an ce o f the opti ma l AAP C wi ll in e vit a bl y de via te fro m t he re f er en ce . Des p it e pe rf or ma nc e de gr ad at i on, th e c om pa ri s on in F ig . 12 i ndi c at es t ha t th e op ti m al A APC i s st i ll s ig ni fi ca n tl y be tt er th a n th e cl ass i c VIC un de r se ve re sce na ri os . T he re fo re , th e opt im al AA PC o f th e W T is e ve nt -i ns e ns it iv e o ve r a rel at i ve ly w ide r a nge of s cen ari o s. To qu an ti t at e th e per f orm a nce de g rad at i on of the op ti ma l AA PC , t he i n di cat o r is d e fi ne d a s fol l ows . % nadir nadir r nadir ff e f    −  =  ( 40 ) In th is pap er , th e eve nt -i ns e ns it iv e ra ng e is ad opt e d as 5% r e  . Un de r th is c ri te ri o n, Fi g. 13 s hows the im pa ct o f t he wi n d s pe ed of th e WT on the ev ent - i ns ens i ti ve ran g e of the opt i ma l AAP C, whe r e 9 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < max d P d e no te s th e upp e r li mit of th e eve n t-i n se ns it iv e ran g e. Th e re s ul ts i ndi c at e tha t the op ti m al AA PC o f W Ts is ev e nt- i ns en si ti ve to a ge ne ra l m aj o r- pow e r di st u rb an ce su c h as a 10% lo a d s ur ge a t di f fe re nt w i nd s pe ed s. Th e ab o ve re s ult s in di ca t e t h at th e op t ima l AA PC d er iv e d f r om a s pe ci fi e d dis t ur ba nc e is al so op t im al for di ve rse dis t ur ba nc e ma gn i tu des . He nc e, t he o pt im al AA PC o f W Ts c an be s ol ve d and de pl o ye d ba s ed o n t he o n- li n e ro ll i ng up dat e u nd e r a hy p ot he ti ca l di st u rb an ce , a voi d in g t he he a vy p os t- e ve nt c om p uta ti o na l bu r den . VI. C ASE S TUDY II: IEEE 39 - B US S YSTEM Th e IEE E 39 -b us s ys te m sh ow n in Fig . 14 is co ns tr uc te d to ve r if y th e ef fe ct i ve ne ss o f th e pr op os e d AA PC in th e la rg e- sca l e po we r sy st em . S pe ci f ic al ly , equ iv a le nt SG of t he ext e rn al s yst em G1 a nd st eam t ur bi nes G 2/ G5/ G6 /G 7 /G 8/G 9 ar e eq ui pp ed wit h IE E EG 1 ty pe of go ve rn or ; hy dro tur bi nes G3 /G 10 are eq ui pp ed wi t h H YG OV ty pe of go ve rn or ; gas tu rb in e G4 is eq ui ppe d wi th GA S T ty pe of go ve rn or . The de ta il ed mo del o f th e a bov e go ve r no rs can be fo un d in [ 26 ]. In a dd it i on , 5 agg re ga te WTs are co nn e ct ed to th e s yst e m at Bu s 8/ 14 /1 6/ 18 / 2 2, den ot ed as WT1 , W T2 , W T 3 , W T4, an d WT 5 re s pe ct iv el y . Ea c h ag gr eg at ed WT is c om pos e d of 80 5M W  D FI G -ba s ed W Ts , a nd t he w in d s pe ed i s se t t o 6. 5 m/s i n W T1 , 7. 5 m/ s in W T2 , 8. 5m /s in WT 3 , an d 9.5 m/ s in WT4 , an d 10. 5 m/ s in WT5 , to mo del th e di ve rs it y of o pe r at in g c o ndi t io ns of W Ts . Th e mo del and pa r am et er s o f t h e WT a re t h e sam e as i n S ec ti o n V . F irs t , th e tra j ec to ry op ti mi za t io n mo de l is con st r uct e d and s ol ve d, an d th e ag gregated op ti m al AA PC o f all the W Ts can be obt ai ne d , wh i ch ne e ds to be reasonably allocated to each WT. C or re sp on di ng l y, the al lo ca ti on fa ct or s o f WT s are de ri ve d a s 1 0. 08 c = , 2 0. 21 c = , 3 0.3 74 c = , 4 0.2 48 c = , 5 0.0 87 5 c = . Th e pe rf or m anc e of th e pr op os ed op ti ma l AAP C is co mp ar ed wi t h th e fo ll o wi ng t h ree i n ert i al c on t rol s t ra te gi es . S tr at eg y 1 : Cl as s ic al VI C i n [2] , tha t is , th e coe ff i ci ent s of th e pr op or t io n al -d eri v at iv e (PD ) con t ro ll er rem ai n con st a nt. In th is pa pe r , 20 f k = , a nd 10 in k = . S tr at eg y 2 : Ad a pti ve VIC i n [3] . Sp ec if ic al l y, th e c oef f ic ie nt s o f VI C are ada pt iv e ly adj u st e d ac co r di ng t o t he re le as a bl e kin et i c en e rg y of t he W T. S tr at eg y 3 : Op ti m al te mp ora ry fr e qu enc y su pp ort con tr ol (T FS C ) in [1 6] . T he d if f er en ti al c on t ro ll e r i n th e t ra di ti on al V IC i s re p la ce d wi th a fi rs t -o rde r l ow -p as s fi lt e r. Th e n, the f ol lo wi ng tw o t yp ic a l sc e nar i os of pow e r s ho rt age a re de si g ne d to v e ri fy t he pe rf or man c e of t h e ab ov e me th o ds . S cen ar i o 1 : Ac ti v e lo a d of t he s ys te m s ur ge s b y 10% . S cen ar i o 2 : G7 t ri ps . F or a lar ge -s ca l e powe r sy st e m, th e gr id fr e qu enc y i ne vi ta bl y ap pe a rs sp a ti al a nd te m po ral dis t ri bu ti o n di ffe r en ces pos t a ma jo r po we r defi ci t , ta ki ng int o acc ou nt th e osc il l at i on of SG s. Gi ve n th is , a w i del y us ed co nc e pt is in tr od u ce d i n t hi s pa pe r, na m el y th e fr e qu en cy of t he ce nt er o f i ne rt i a (Co I) , whi ch ca n re fl ec t the av e ra ge fr e qu en cy dyn ami cs of th e gri d . Ac co r di ng to [2 7], the fr e qu en cy of t he C oI i s ca lc u la te d as f o ll ow s. g g gi gi i i CoI gi gi i H S f f HS   =   ( 41 ) Fig. 14 Modified I EEE 39-bus system . Fig. 15 Freque ncy dynamics of SGs a nd CoI. (a) Scenario 1. ( b) Scenario 2. F ig . 1 5 s h ows th e fr e que nc y d yna m ic s o f e a ch S Gs a n d t h e C oI in scen ari o 1 a nd s ce na ri o 2 , wi t h WTs equ ip pe d wi t h t he opt ima l AA PC . It can be se en tha t th e fre qu en cy of ea ch S Gs s h ows ce rta i n os ci l la ti o n cha ra ct er is t ic s at the in it ia l sta g e of a po we r dis t ur ba nc e. F or tu na te ly , the fr eq ue nc y of th e C oI eli mi na te s the fr eq ue nc y di f fe re nc e cau se d by the os c il lat o ry be ha vi or o f SG s and pr op er ly de pi ct s the ave ra ge gri d fre qu enc y dyn am ic s, as sh ow n in the r ed s oli d li ne in Fi g . 15 . Hen ce , it is re as on ab le to mo de l th e ov er al l gr i d dy na mi cs by th e fr eq ue nc y of th e CoI . If th e re is no s pec ia l de s cr ip ti o n, t he fr e qu en cy i n the fo l lo wi n g ref er s to t ha t of t he Co I. 10 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < Fig. 16 Comparison of system frequency dynamics under different control strategies. (a) Lo ad surge. (b) Ge nerator trippin g. TABLE III C OMPARISON OF F REQUENCY N ADIR I N S CENARIO 1&2 (Hz) nadir f  Scenario 1 Scenario 2 No FR -0.4133 -0.3304 Strategy 1 -0.3193 -0.2630 Strategy 2 -0.2936 -0.2410 Strategy 3 -0.2950 -0.2515 Proposed AAPC -0.2458 -0.2068 Mo r eo ve r, it s ho ul d b e n ot ed t hat t he f req u en cy of th e C oI is on ly us e d f or r es ul t d is pl ay . Th e d ep en de nt fr eq ue nc y of th e im p le me nt at io n o f t h e o pti ma l A APC is s t il l me as ur e d l o ca ll y. Th e gr i d fre q ue nc y dyn am ic s wi t h th e pro po se d c on tr ol s tr at e gy an d S t rat eg i es 1-3 in S ce na ri os 1&2 ar e c om pa re d i n Fi g. 16 , and th e de v ia ti o n o f fr e qu enc y n ad ir s in t he t r ans i en t pr oc es s ar e li s te d in TA BL E III , wh er e “N o FR ” de no t es tha t WT s do not pr ovi de fr e qu en cy s up por t . The re su lt s in TAB LE III i ndi ca t e tha t the te m po ra r y FR of WTs does si gn if i ca ntl y im pr ov e the fr eq ue nc y na di r of th e po we r sy st e m pos t a ma jo r po we r de fi c it . Ta ki ng S cen a ri o 1 as an ex am pl e, the fre q ue nc y nadi r dev ia ti o n wit h WT s un de r the ab ov e co nt ro l s t rat e gie s is imp r ove d by 22 .7 4 % ( S t ra te gy 1 ), 28 .9 6% ( S t ra te g y 2 ) , 28 . 62% ( S t ra te gy 3 ), 40 .5 3 % ( o p ti ma l AA PC ) res pe ct i vel y, co m pa re d wi th “N o FR ” . F ur th er mo re , th e co mp ar is on res ul ts i ndi c at e tha t the pr op os e d op t im al AAP C is ef fe ct i ve in a la rg e-s ca l e p owe r sys te m wit h mu l ti pl e W Ts , wh ic h c on tr ib ut es to a hi gh er f re qu en cy na di r co mp a re d wi t h t he e xis t in g c on tr ol st rat e gi es . Sp ec i fi ca ll y , co mp a re d wi th s t ra te gi es 1- 3, t he o pt im a l AAP C of WTs im pr ove s th e fre q ue ncy na di r de v ia ti on i n t he ab ove tw o sc e nar i os by 23 .0 2% a nd 2 1. 3 7% ( St r at eg y 1 ), 16 .2 8% an d 1 4. 19 % ( S t ra teg y 2 ), 16 . 68 % an d 17 . 78% ( S tr at egy 3 ) , re sp e ct iv el y. To de mo ns t ra te th e e ff ec ti ve ne ss of th e pro po se d al lo c at io n s tr at e gy amo ng WT s, it is co mp ar e d wit h th e av er age al lo cat i on pr i nc i ple . The dy nam i cs of the sys te m fre qu e nc y an d WT s wit h the Fig. 17 Dynamics of WTs unde r different allocation strategies. (a) Output power of WT5 u nder h igh wind spe ed. (b) Rotor speed of WT1 u nder low wi nd speed. (c) System fr equency. pr op os ed all oc at io n st rat e gy and av e ra ge al l oca ti o n i n S ce na rio 1 ar e com pa r ed as s how n in F ig. 17 . The res ul ts s how th at the pr op os ed all o ca ti o n st rat e gy tak es in to a cc ou nt the di ff er en ce s in op er a ti ng co nd it i ons and po we r sup po r t mar gi ns o f WT s . Th e re as o na bl e all oc at io n fac to r ef fec t iv el y av o ids t h e ove r -l ow er -l im i t of th e rot or sp ee d of the WT w ith lo w win d spe e d ( WT 1), as sho wn in Fi g . 17 ( b) ; a s wel l as avo i ds the o ve r- up pe r li m its of t he ou tpu t po we r of th e WT wit h hig h win d s pe e d ( WT 5), as s how n in Fi g. 17 (a ) . Ov er -l ow er lim it s ma y ca us e co nc er ns a bo ut the saf e op er a ti on o f th e WT . As f or th e ov er -u pp e r-l i mi t of t he powe r re f er en ce , alt h ou gh t he li mi t er en s ure s a bea ra bl e ac tu al o utp u t po we r of th e WF , th e de vi at io n of th e out pu t po we r fr om the ex pe c te d val u e wil l i nev it a bl y r es ul t in a de gr ad at io n in c ont ro l pe r fo rm an ce , i.e . de te ri o ra ti ng th e fr eq ue nc y nad i r, as s how n in F ig . 17 (c) . The ab ov e res ul ts sh ow t hat the pr op os ed al lo ca ti on s tr at e gy a mo ng WT s i mp ro ve s t he tr a ns ie nt fr eq ue nc y s ta bi li ty of th e p ow er s ys t em w hi le e ns u ri ng t he s a fe o pe ra ti on of t he W T. VII. C ONCLUSION In this paper, a system-view optimal AAPC of WTs i s proposed to provide fast frequency support for power systems . Different f rom the widely used VIC methods, this paper explores the optimal frequency sup port mode of WTs based on the system dem ands. As a feed back approximation , the proposed AAPC of WTs inher its th e op timal performance o n maximizing the frequen cy nadir of the trajectory op timization model. More imp ortantly, the univer sality of the reverse approximation is strictly proved. Besides, the event insensitivity of the proposed method avoids the depend ence on the disturbance magnitude. Hence, the optimization calculation can b e preposed by rolling implementation to avoid the heavy 11 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < post-even t co mmunication and computation burden . Extensive simulation r esults prove that th e propo sed meth od can effectively improve the sy stem frequency stability post a major power d eficit. A PPENDIX A D YNAMI C OF THE G OVERNOR S YSTEM In the ASF model, the d ynamics o f separate governor is retained, and th e FR can be expressed as ( ) ( ) ( ) mi gi P s G s f s  =  ( 42 ) where g i  . Ignoring the nonlinearity of governors, the following state - space form of FR o f governors can be derived from ( 42 ). ( ) ( ) ( ) ( ) ( ) ( ) g i gi gi gi mi gi gi gi t t f t P t t D f t  =  +      =  +    x A x B Cx ( 43 ) where g i  ; 1 i m gi   x are the state variables o f th e i -th governor; ii mm gi   A , 1 i m gi   B , 1 i m gi   C , an d gi D  are the coefficien t matrices; i m is the order s of the governor. Then, the overall dynamics of the governor system can be expressed as fo llows. ( ) ( ) ( ) ( ) ( ) ( ) g g g g g g g g t t f t P t t D f t  =  +      =  +    x A x B Cx ( 44 ) where 1 2 g g g gM    =     x x x x , 1 2 g g g gM    =     A A A A , 1 2 g g g gM    =     B B B B ,   12 g g g g M = C C C C , g g gi i DD  =  A PPENDIX B P RINCIPLE OF THE G AUSS P SEUDOSPECTRAL M ETHO D In the Gauss pseudospectral method, the interpolation poin ts are discretely distrib uted in the time interval of [ 1 , 1 ]  − . Hence, the time interval needs to be converted into 0 00 2 f ff tt t t t t t  + =− −− ( 45 ) Taking the roots of the K- order Legendre polynomial as the discrete p oints, den oted as 1 2 {} K    = , the K-order Lagrangian inter polation of the state variable is as fo llows. 0 ( ) ( ) ( ) ( ) K ii i x X L x     = =  ( 46 ) where 0, () K j i j j i ij L    = − = − is the basic fun ction of Lagrang ian interpolation . Similarly, the co ntrol variables should also b e discretized. 0 ( ) ( ) ( ) ( ) K ii i u U L u     = =  ( 47 ) Notably, the e ndpoint of the interval 1 f  =− is not involved in the above interpolation points, whereas constraints related to the final valu e of the states are co mmon. Therefore, an estimation for f X needs to b e co nstructed by the Gau ss integration algorithm. First, the theoretical final v alue of the state can be der ived from the dynamics of the system. 1 0 1 ( ) ( ) ( ( ), ( ), ) f x x f x u d       − =+  ( 48 ) Then, its discrete estimation can be calculated by 0 00 1 ( ) ( ) ( ( ), ( ), ; , ) 2 K f f k k k f k X X f X U t t        = − =+  ( 49 ) where k  is Gaussian weig ht. Then, th e differential of the state variable is ap proximated by interpolation , as follows. 00 ( ) ( ) ( ) ( ) ( ) K K k k i k i ki i ii x X L X D X      ==  = =   ( 50 ) where ( 1 ) KK +  D is the differential matrix formed by the differential of th e basic functions of Lagrangian inter polation. Finally, the differ ential eq uations of the dynamic system can be converted to the following algebraic equ ations. 0 0 0 ( ) ( ( ), ( ), ; , ) 0 2 K f ki i k k f i tt D X f X U t t     = − −=  ( 51 ) The continuous p ath constraints are also discretized into the following form . 0 ( ( ), ( ), ; , ) 0 k k k f C X U t t     ( 52 ) So far, th e inf inite-dimensional con tinuous trajectory optimization problem has been tr ansformed in to a nonlinear programming (NLP) problem, for which there are many efficient solvers, such as the sequential quadratic programming (SQP) method [28]. A PPENDIX C A LLOCATION S TRATEGY OF M U LTI - WT S To quan tify the frequ ency support ca pability o f the WT, the following two indices are in troduced, deno ting its maximum releasable kinetic en ergy and the m aximum increasable active power. max min 0 max ma x 0 , kj kj kj w ej ej e j E E E j P P P   = −     = −   ( 53 ) where 2 0.5 kr EJ  = is th e k inetic energy o f th e WT. The subscript “0”, “min”, and “max” denote the initial v alue, minimum v alue, and maximum value, respectiv ely. Larger max kj E  and max ej P  indicates a stronger frequ ency support cap acity, but they are irr econcilable, as shown in Fig. 5. It can be seen that suf ficient kinetic en ergy means that the WT operates at a h igh wind speed, whereas the increasable power is limited for the lar ge steady -state po wer, and vice versa. Therefore, the allocation factor of WTs is designed as max max max max min{ , }, ww kj ej j w kj ej j j EP cj EP     =   ( 54 a) , w j j w j j c cj c    =  ( 54 b) 12 > REPLACE THIS LINE WI TH YOUR MANUS CRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT ) < R EFERENCES [1] G. W. E. C, “Global Wind Report 2022,” Global Wind Energy Council, Brussels, Belgi um, Tech. Rep., 20 22. [2] J. Morren, S. W. H. de Haan, W. L. Kling and J. A. Ferreira, “Wind turbines emulating inertia and supporting primary frequency control,” IEEE Trans. Power Syst. , vol. 2 1, no. 1, pp. 433-43 4, Feb. 2006. [3] J. Lee, E. Muljad i, P. Srense n , and Y. C. Kan g, “Releasable kinetic energy based inertial control of a DFIG wind power plant,” IEEE Trans. Sustain. Energy , vol. 7, no. 1, pp. 279 – 288, Ja n. 2016. [4] X. Lyu, Y. Jia, and Z. Dong, “Adaptive frequency responsive control for wind farm consideri ng wake interaction,” J. Modern Power Syst. Clean Energy , vol. 9, no. 5, pp. 1066 – 1075, Sep. 2021. [5] M. Toulabi , A. S. Dobak hshari, an d A. M. Ranjbar, “An a daptive feedback linearization approach to inertial frequency response of wind turbines,” IEEE Trans. Sustai n. Energy , vol. 8, no. 3, pp. 916 – 926, Jul. 2017. [6] Q. Jiang et al., “Time -Sharing frequency coordinated control strategy for PMSG- Based wi nd turbine,” IEEE J. Emerg. Sel. Topics C irc uits Syst. , vol. 12, no. 1, pp. 2 68 – 278, Mar. 2022. [7] A. Ashouri-Zade h and M. Toulabi, “Adapt ive v irtual inertia controller for DFIGs considering nonlinear aerodynamic efficiency,” IEEE Trans. Sustain. Energy , v ol. 12, no. 2, pp. 1 060 – 1067, Apr. 2 021. [8] D. Yang, G. - G. Yan, T. Zheng, X. Zhang and L. Hua, “Fast Frequency Respons e of a DFIG Ba sed on Variable Power Point Tracking Control,” IEEE Trans. Ind. Appl. , vol. 58, no. 4, pp. 5127-5135, July-Aug. 2 022. [9] S. Huang, Q. Wu, W. Bao, N. D. Hatziargyriou, L. Ding, and F. Rong,“Hierarch ical optimal control for synthetic inertial resp onse of wind farmbased on alternating direction method o f multipl iers,” IEEE Trans. Sustain. Energy , v ol. 12, no. 1, pp. 2 5 – 35, Jan. 2021. [10] W. Bao, Q. Wu, L. Di ng, S. H uang, and V. Terz ija, “A hierarchical inertia l control scheme for multiple wind farms wit h BESSs based on ADMM,” IEEE Trans. Sus tain. Energy , vol. 12, no. 2, pp. 1461 – 147 2, Apr. 2021. [11] L. Guo et al., “Do uble -Layer Feedback Control Method for Synchronized Frequency R egulation of PMSG- Base d Wind Farm, ” IEEE Tr ans. Susta in. Energy , vol. 12, n o. 4, pp. 2423-2435, Oct. 2021. [12] Z. Guo and W. Wu, “Data -Driven Model Predictive Control M ethod for Wind Farms t o Provide Frequency S upport,” IEEE Trans. Ener gy Convers. , vol. 37, no. 2, pp. 1304-1313, June 20 22. [13] Z. Wang and W. Wu, “Coor dinated control method for DFIG -based wind farm to p ro vide pri mary f requency regulation service,” I EEE T rans. Power Syst. , vol. 33, no. 3, pp. 2644 – 2659, Ma y 2018. [14] A. S. Ahmadyar and G. Verbiˇc, “Coordinated operation strategy of wind farms for frequency control by exploring wake interaction,” IEEE Trans. Sustain. Energy , v ol. 8, no. 1, pp. 23 0 – 238, Jan. 2017. [15] S. Ma, H. Geng, G. Yang, and B. C. Pal, “Clustering -based coordinated control of la rge- scale wind farm for po wer system frequency support, ” IEEE Trans. Sus tain. Energy , vol. 9, no. 4, pp. 1555 – 1564, Oct. 2 018. [16] M. Sun et al., “Novel Temporary Frequency Support Control Strategy o f Wind Turbine Gen erator Considering Coord ination With Synchr onous Generator,” IEEE Trans. S ustain. Energy , v ol. 13, n o. 2, pp. 1011-1020, April 2022. [17] P. Kou, D. Liang, L. Yu, and L. Gao, “Nonlinear model predi ctive control of wi nd f arm for system frequ ency su pport,” IEEE Trans. Power Syst. , vol. 34, no. 5, pp. 3 547 – 3561, Sep. 2019. [18] Z. Chu, U. Markovic, G. Hu g, and F. Teng, “ Towards o ptimal system scheduling with s ynthetic inertia provision from w ind turbines,” IE EE Trans. Power Sys t. , vol. 35, no. 5, pp. 4056 – 4066, Sep. 202 0. [19] E. Riquelme, H. Chavez and K. A. Barbosa, “RoCoF - Minimizi ng H₂ N orm Control Strategy for Multi- Wind Turbine Synthet ic Inertia,” IEEE Acc ess , vol. 10, pp. 18268-1 8278, 2022. [20] M. L. Chan, R. D. Dunlop, and F. Schweppe, “Dynamic equivalents for average system frequency behavior following major distribances,” IEEE Trans. Power Ap p. Syst. , vol. 91, no. 4, pp. 1637 – 1642, Jul. 1 972. [21] P. M. Anderson and M . M irhey dar, “A low-order sy stem frequency response model,” IEEE Trans. P ower Syst. , vol. 5, no. 3, pp. 720-729, Aug. 1990 [22] W. H e, X. Y uan, and J. H u, “I nertia provisio n an d estima tion of P LL -based DFIG wind turbines,” I EEE Trans. Power S yst. , vol. 32, no. 1, pp. 510 – 521, Jan. 2017. [23] F. L. Lewis, D. Vrabie, and V. S yrmos, O ptimal Control , 3 rd ed. N ew York: Wiley, 2012. [24] D. A. Ben son, “A g auss pseudos pectral transcription for opti mal control,” Ph.D. dissertation, Dept. Aeronaut. Astronaut., Mass. Inst. Technol., Cambridge, No v. 2004. [25] A. D. Hansen, F. Lov, and P. Sø r ensen, N. Cutululis, C. Jauch, F . Blaabjer g . “Dynamic wind turbine models in power system s imulation tool digsilent,” Risø Na tional Laborator y, Roskilde, Denmar k, Aug. 2007. [26] P. Po urbeik, “Dynamic models f or turbine -gover nors i n power s ystem studies,” IEEE PE S Tech. Rep. PE S-TR1 , 2013. [27] S. Azizi, M . Sun, G. Liu, and V. Terzija, “Local freque ncy -based estimation of the r ate of c hange of fr equency of the center of i nertia,” IEEE Trans. Power Sys t. , vol. 35, no. 6, pp. 4948 – 4951, Nov. 2 020. [28] P. T. Boggs and J. W. Tolle, “Sequential q uadratic programming,” Acta numerica, vol. 4, pp. 1 – 51, 1995. Yu bo Zhang. (S ’ 21 ) received the B.S. degree in electrical engineering from Xi’an Jiaotong University, China, in 2019. He is currently pursuing the Ph.D. degree in the schoo l of Electrical En gineering in Xi’an Jiao tong Un iversity. His main field of interest includes the control for renewable energy, an d power sy stem frequency stability . Zhiguo Hao . ( M’10 ) was born in Ordos, China, in 1976. He received his B.Sc. and Ph.D. degrees in electr ical engin eering from Xi'an Jiaotong University, Xi'an, China, in 1 998 and 2007, respectively. He has been a Professor with the Electr ical Engineering Depar tment, Xi'an Jiaoto ng University. His resear ch interest includes power system protection and control. So nghao Yang. ( S’18 - M’19 ) was born in Shandong, China, in 1989. He receiv ed the B.S. and Ph.D. degrees in electrical engineering fr om th e Xi’an Jiaotong University, Xi’an , China, in 201 2 and 2019, respectively. Besides, he receiv ed the Ph.D. degree in electrical and electronic engineer ing from T okushima University, Japan , in 2019. Currently, he is an A ssociate Professor at Xi’an Jiaotong University. His research interest includes power system control and protection. Baohui Z hang. ( SM‘99 - ’F’19 ) was born in Hebei Province, China, i n 1953. He received the M.Eng. and Ph.D. degrees in electrical en gineering from Xi’an Jiaotong University, Xi’an, China, in 1982 and 1 988, respectively. He has been a Professor in the Electrical Enginee ring Department at Xi’an Jiaoto ng University since 1992 . His resear ch interests are system analysis, con trol, communication, and protection.